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Theory and Applications of Categories, Vol. 8, No. 17, 2001, pp. 465–480.
ESSENTIAL LOCALIZATIONS AND INFINITARY EXACT
COMPLETION
ENRICO M. VITALE
ABSTRACT. We prove the universal property of the infinitary exact completion of a
category with weak small limits. As an application, we slightly weaken the conditions
characterizing essential localizations of varieties (in particular, of module categories) and
of presheaf categories.
Introduction
An essential localization is a reflective subcategory such that the reflector has a left
adjoint. In [12], Roos gave an abstract characterization of essential localizations of module
categories, proving that they are those complete and cocomplete abelian categories with
a regular generator, satisfying the following conditions (we write the conditions in a nonabelian style, more convenient for the general framework of this work)
(AB4*) Regular epimorphisms are product-stable ;
(AB5) Filtered colimits are exact, i.e. commute with finite limits ;
(AB6) Given a small family of functors (Hi : Ai → A)I defined on small filtered categories,
the canonical comparison τ is an isomorphism
τ : colim( Ai →
A → A) −→
(colimHi ) .
I
I
I
In a subsequent paper [13], Roos introduced a weaker form of (AB6) :
(WAB6) With the same notations as in (AB6), the comparison morphism τ is a regular
epimorphism.
It is then natural to look for a representation of the abelian categories satisfying the same
list of conditions as in Roos’s theorem, but replacing (AB6) with (WAB6).
The first aim of this paper is to prove that this set of conditions, which seems weaker,
still characterizes essential localizations of module categories. In fact we prove a more
general result : Roos’s theorem has recently been generalized by Ad´amek, Rosick´
y and
the author to a non-additive context [1]. They have characterized essential localizations of
(multi-sorted, finitary) varieties as those complete and cocomplete Barr-exact categories
Received by the editors 2000 November 7 and, in revised form, 2001 August 24.
Transmitted by W. Tholen. Published on 2001 September 5.
2000 Mathematics Subject Classification: 18A35, 18B15, 18C05, 18E10, 18E35, 18F20.
c Enrico M. Vitale, 2001. Permission to copy for private use granted.
465
Theory and Applications of Categories, Vol. 8, No. 17
466
with a small regular generator, satisfying (AB4*), (AB5) and (AB6). We show that in this
characterization (AB6) can be replaced by (WAB6). Since abelian varieties are exactly
module categories, as a particular case we obtain the new formulation of Roos’s theorem.
A similar analysis can be done for Roos’s characterization of essential Grothendieck
topoi (i.e. essential localizations of presheaf categories) [10, 11] (see also Section 4 in
[1] for a different approach). In fact we prove that the complete distributivity involved in
Roos’s theorem can be weakened : instead of an isomorphism, the comparison morphism
can be assumed to be a regular epimorphism.
To obtain our results, we use in a systematic way the exact completion of a category
with weak limits [5, 6]. This uniform approach stresses one more time the striking analogy
between the varietal context and the presheaf context: not only the results are similar,
but also the proof technique is essentially the same.
In dealing with essential localizations, we need the infinitary extension of the exact
completion. This has been studied in [6]. Unfortunately, the proof of the main theorem in
[6] is not complete. The claim that a left exact functor which is left covering with respect
to small products is continuous, used to end the proof of Theorem 3.4, is not true, as we
show with a counterexample. The second aim of this paper is then to give a complete
proof of the universal property of the infinitary exact completion.
I would like to thank P.T. Johnstone for his help concerning Example 1.8.
1. The infinitary exact completion
For basic definitions and results on regular and exact categories, the reader can see [2, 5].
We assume familiarity with the exact completion as described in [5] and with the theory
of left covering functors developed therein. Left covering functors are called flat functors
in [6]. We write
Γ : C → Cex
for the exact completion of a weakly left exact category C. As far as the infinitary completion is concerned, we will deal only with arbitrary small limits ; in fact the ranked case
runs exactly as the general case.
A main difference between limits and weak limits is that in the weak case the usual
reduction to small products and equalizers of pairs of parallel arrows does not work. One
has to use small products and equalizers of small families of parallel arrows (we call them
small equalizers). The next three lemmas are the infinitary version of Proposition 1,
Lemma 28 and Proposition 27 in [5].
1.1. Lemma. If a category C has weak small products and weak small equalizers, then it
has weak small limits.
Proof. The proof is similar to that of the finitary case. We give some details in view of
Lemma 1.3.
I) Consider a small diagram of arrows with the same codomain (fi : Xi → X)I , a weak
Theory and Applications of Categories, Vol. 8, No. 17
product (πi :
I
467
Xi → Xi )I and a weak small equalizer
e
E
/
πi ·fi
:
I Xi
/X ;
then the cone (e · πi : E → Xi )I is a weak limit over (fi : Xi → X)I (a weak small
pullback).
II) Let
L : D → C be a functor with D a small category ; consider a weak product
(πD : D L(D) → L(D))D and, for each arrow d : D → D′ in D, a weak equalizer
Ed
ed
πD′
/
/
D L(D)
πD ·L(d)
′
/ L(D ) ;
consider also a weak small pullback (ǫd : E → Ed )d on the diagram (ed : Ed →
Then (ǫd · ed · πD : E → L(D))D is a weak limit of L : D → C.
D
L(D))d .
Recall that a category B is completely regular if it is regular, complete and regular
epimorphisms are stable under small products (a condition which is redundant in the
finitary case).
1.2. Lemma. Let B be a completely regular category.
1) If in the following commutative diagram, where the horizontal arrows represent small
equalizers, f1 is a regular epimorphism and f2 is a mono, then the unique factorization f is a regular epimorphism
E
e
/ A1
ai
:
/ A2
f1
f
L
/ B1
l
f2
bi
:
/ B2
2) Consider two naturally connected small families of arrows with common codomain
(that is ai · f = fi · bi for each i in a small set I)
Ai
ai
/A
fi
f
Bi
bi
/B
and the unique factorization ϕ : P → Q between the corresponding small pullbacks ;
if f is a mono and each fi is a regular epimorphism, then ϕ is a regular epimorphism.
1.3. Lemma. Let C be a category with weak small limits and B a completely regular
category. If a functor F : C → B is left covering w.r.t. weak small products and weak
small equalizers, then it is left covering w.r.t. weak small limits.
Theory and Applications of Categories, Vol. 8, No. 17
468
Proof. I) F is left covering w.r.t. weak small pullbacks : consider a small family of
convergent arrows in C and a weak pullback as described in the first step of Lemma 1.1
e
E
/
Applying F we obtain
Fe
F (E)D
L
DD
DD
t DDD
"
zS
q zzz
z
zz
|z
z
/ F(
:
uu
uu
u
u
uu s
uu
/
πi
I Xi
/ Xi
fi
/X
F (πi ·fi )
I Xi )
/ F (X)
p
1
I F (Xi )
pi
/ F (Xi )
F fi
/ F (X)
where s is the equalizer of the F (πi ·fi )’s and the bottom line is the pullback of the F (fi )’s
computed using products and equalizers in B. By assumption, p is a regular epi, so that
by Lemma 1.2 also q is a regular epi. Also t is a regular epi by assumption, so that the
comparison t · q : F (E) → L is a regular epi.
II) F is left covering w.r.t. weak small limits : consider a functor L : D → C with D
small, and a weak limit as in the second step of Lemma 1.1
ǫd
πD
/ Ed ed /
/ L(D) .
E
D L(D)
In the same way we build up the limit of L · F : D → C → B :
L
λd
/ Ld ld
/
D F (LD)
π
˜D
/ F (LD) .
Let us call l = λd · ld : L → D F (LD) ; it is the limit of
π
˜D ·F (Ld)
/
′
/ F (LD )
D F (LD)
π
˜D′
d
and then it is a mono. Now consider the pullback
l /
LO
D F O(LD)
p
p′
P
l′
/ F(
D LD)
′
By assumption,
p is a regular epi and then also p is a regular epi. Moreover, the mono
′
l : P → F ( D LD) is the limit of
F (πD )·F (Ld)
/
′
F ( D LD)
/ F (LD )
F (πD′ )
d
Theory and Applications of Categories, Vol. 8, No. 17
469
But this limit can be obtained in two steps : first taking for each d the equalizer
Sd
sd
F (πD )·F (Ld)
/ F(
D LD)
F (πD′ )
/
′
/ F (LD )
andthen taking the pullback (σd : P → Sd )d of the family of convergent arrows (sd : Sd →
F ( D LD))d . By assumption, the factorization qd : F (Ed ) → Sd is a regular epi ; then,
by Lemma 1.2, the factorization q : Q → P is a regular epi,
where (jd : Q → F (Ed ))d
is the pullback of the convergent family (F (ed ) : F (Ed ) → F ( D LD))d . Moreover, by
the first part of this lemma, the factorization t : F (E) → Q is a regular epi. Finally, the
comparison
F (E)
t
q
/Q
p′
/P
/L
is a regular epi because each part is a regular epi.
The fact that, if C has weak small limits, then Cex is completely regular, is stated
without proof in [5]. In the next lemma, we add an explicit description of small products
in Cex . This description will be used in the proof of Proposition 1.6.
1.4. Lemma. Let C be a category with weak small limits. Its exact completion Cex is
completely regular and Γ : C → Cex is left covering w.r.t. weak small limits.
Proof. The last fact follows from the fact that C is a projective cover of the complete
category Cex . The fact that in Cex regular epis are product-stable follows from the fact
that they are of the form
/S
R
X
1
/X
and from the description of products in Cex given hereunder.
Consider a family of objects
r0i
in Cex ; their product is given by
Ri
r1i
ei
E
e0
e1
I Xi πi
/
/ Xi
I
/ Ri
r0i
/ Xi
r1i
Theory and Applications of Categories, Vol. 8, No. 17
470
where πi : I Xi → Xi is a weak product in C and E is a weak limit in C as in the
following diagram
E
x FF
xx
xx
x
x| x
e0
I Xi
ei
πi
Xi o
FF e1
FF
FF
"
Ri
r0i
I Xi
πi
/ Xi
r1i
The next lemma generalizes Proposition 20 in [5]. Since the proof runs as in the
finitary case, we omit it.
1.5. Lemma. Let F : A → B be a functor between complete categories. If F is left
covering w.r.t. small limits, then it is continuous.
The next proposition is the main result of this section. It completes the preparatory
work to establish the universal property of the infinitary exact completion. In the proof,
we give an explicit description of small equalizers in Cex .
1.6. Proposition. Let C be a category with weak small limits, B an exact and completely
regular category and F : C → B a functor left covering w.r.t. weak small limits. Then the
exact extension Fˆ : Cex → B is continuous.
Proof. Recall that Fˆ sends an object of Cex , that is a pseudo equivalence relation
/
r0 , r 1 : R
/ X , to the coequalizer of the pair (F r0 , F r1 ). This coequalizer exists because, by Theorem 26 in [5], the jointly monic part of the regular epi-jointly mono factorization of (F r0 , F r1 ) is an equivalence relation. Fˆ extends to morphisms via the universal
property of the coequalizer. By Lemma 1.5, to prove that Fˆ is continuous is enough to
prove that Fˆ is left covering w.r.t. small limits. By Lemma 1.3, it suffices to prove that
Fˆ is left covering w.r.t. small products and small equalizers.
I) Consider a small product in Cex
ei
E
e0
/ Ri
r1i
r0i
e1
I Xi πi
/ Xi
Using F and Fˆ we obtain the following commutative diagram
F ( I Xi )
λ
q
/
Fˆ (e0 , e1 )
µ
/
I F (Xi )
I qi
ˆ i i
I F (r0 , r1 )
Theory and Applications of Categories, Vol. 8, No. 17
471
Each qi is a regular epi, so that I qi is a regular epi because B is completely regular ; by
assumption λ is a regular epi, so that also µ is a regular epi.
II) Small equalizers in Cex : let
fi
:
R
r0
s0
r1
X
/S
s1
fi
:
/Y
be a small family of parallel arrows in Cex and consider the diagram which contains all
the diagrams of the form
~~
~~
~
~ ~
S
s0
Y
~>
~~
~
~~
~~
s1
fi
X
/Y
fj
for each i = j, all the subdiagrams being connected by the identity on X. Take now a
weak limit E of such diagram
ϕi,j
E
s1
Y
~>
~
~~
~~
~~
e
/S
~
~
s0 ~
~~
~~
fi
X
/Y
fj
and a weak limit R as in the diagram below
e0
Eo
e1
R
e
/E
e
e
Xo
R
r0
/X
r1
The equalizer in Cex is
e
R
e0
r0
e1
E
/R
r1
e
/X
Theory and Applications of Categories, Vol. 8, No. 17
472
III) Consider a small equalizer in Cex as in the previous step
e
R
e0
r0
e1
E
fi
:
/R
e
s0
r1
/X
/S
fi
:
s1
/Y
and apply Fˆ : Cex → B. We obtain
FR
F e0
Fe
F e1
F r1
F r0
FE
F fi
:
/ FR
F fi
:
/ FX
Fe
/ FS
KK
KK p
K
F s1 KKK
F s0
K
n0 K%
o
/ FY
N (q3 )
o
n1
q2
q1
q3
ˆ
/ Fˆ (r , r ) F [fi ] / Fˆ (s , s )
0 1
0 1
:
q8
q
q
qq
t
qqq
qqqq h
Fˆ (e0 , e1 )
Fˆ [e]
L
where the triangle on the right is the regular epi-jointly mono factorization of (F s0 , F s1 )
and the triangle on the bottom is the factorization of Fˆ [e] through the equalizer of the
Fˆ [fi ]’s. We have to prove that t : Fˆ (e0 , e1 ) → L is a regular epi. For this, consider the
limit A in B of the following diagram (a subdiagram for each i = j, all the subdiagrams
connected by the identity of F (X))
ki,j
A
i
/ N (q3 )
w
n0 ww w
w
ww
w{ w
n1
FY
x;
xx
x
xx
xx
F fi
FX
F fj
/ FY
Using the morphisms ϕi,j : E → S, we obtain a factorization σ : F E → A making commutative the following (solid) diagram
σ
FE
/A
q1
i
Fˆ (e0 , e1 )
τ
FX
q2
t
L
h
/ Fˆ (r , r )
0 1
Theory and Applications of Categories, Vol. 8, No. 17
473
Observe that h is a mono, and assume that σ is a regular epi (we will prove this later),
so that there exists τ : A → L making commutative the two triangles. If we can show
that τ is a regular epi, we have that also t is a regular epi, as we need. To show that τ is
a regular epi, we can check that it is the pullback of q3 along h. This can be done using
that (n0 , n1 ) is the kernel pair of its coequalizer q3 (because B is exact) and i is a mono
(because (n0 , n1 ) are jointly monic). It remains to prove that σ : F E → A is a regular epi.
For this, consider the limit A in B of the following diagram (with the usual conventions)
Ki,j
A
i
FY
x;
xx
x
xx
xx
/ FS
y
y
F s0 yy
y
yy
|yy
F s1
F fi
FX
F fj
/ FY
Since F : C → B is left covering, the unique factorization α : F E → A such that α · i = F e
and α · Ki,j = F ϕi,j is a regular epi. Moreover, comparing the construction of A and of A,
we see that the unique factorization m : A → A such that m · i = i and m · ki,j = Ki,j · p, is
a regular epi (use part 2 of Lemma 1.2). Finally, composing with the mono i : A → F X,
we check that σ = α · m.
1.7. Corollary. Let C be a category with weak small limits and consider its exact
completion
Γ : C → Cex .
For each exact and completely regular category B, the composition with Γ induces an
equivalence between the category of exact and continuous functors from Cex to B and the
category of functors from C to B left covering w.r.t. small limits.
Bearing in mind the fact that a left exact functor which preserves small products is
continuous, and that a functor between complete categories is left covering w.r.t. small
limits iff it is continuous (Lemma 1.5), one could think that a left exact functor which is
left covering w.r.t. small products is indeed continuous. This is wrong. In fact, let C and
B be as in Corollary 1.7 and assume F : C → B be left covering w.r.t. weak finite limits
and weak small products. A glance at the proof of Proposition 1.6 shows that the exact
extension Fˆ : Cex → B is (left exact and) left covering w.r.t. small products. Now, if Fˆ is
continuous, then F is left covering w.r.t. weak small limits. The following example shows
that this is not always true.
1.8. Example. Let C be a small category and let i : B → [C op , Set] be a persistent
localization of the presheaf category [C op , Set], i.e. the left adjoint F : [C op , Set] → B is left
exact and units are regular epis (see [7]). First of all, observe that B, being a Grothendick
topos, is exact and extensive and has a family of regular generators. Moreover, since units
are regular epis, the generators are regular projective and indecomposable objects (see
Theory and Applications of Categories, Vol. 8, No. 17
474
[9]). Then B is equivalent to a presheaf category and then (as any exact and complete
category with enough regular projective objects) it is completely regular. Let us now
prove that the reflector is left covering w.r.t. small products. Consider a family of objects
op
in [C op , Set]
(Xj )J
and their product in [C , Set] and in B. Consider also the comparison
e : F ( J Xj ) → J F (Xj ) ; we have to show that e is a regular epi. Consider the following
commutative diagram
i(e)
i(F ( J Xj ))
fMMM
MMM
M
η J Xj MMM
J Xj
/ i(
J F (X5 j ))
≃
k
kkkk
k
k
k
kkkk J ηXj
kkkk
J i(F (Xj ))
(where ηX : X → i(F (X)) is the unit at the point X). Since units are regular epis and
[C op , Set] is completely regular, i(e) is a regular epi. But, since i is full and faithful and left
exact, it reflects regular epis, and then e is a regular epi. To finish the argument, we need
an example of a persistent localization of a presheaf category such that the reflector is not
continuous. Such an example is provided by Example 2.6 in [7] : let M be a semilattice
with a non-principal ideal I and take as B the full subcategory of the category M -Set of
those objects on which the elements of I act trivially. B is a persistent localization, but
the reflector does not preserve small products, as communicated to me by P.T. Johnstone.
2. Essential localizations of varieties
Let B be a complete and cocomplete category and consider the following conditions:
(LM) B is exact and has a small regular generator G. This means that G = (Gj )J is a
small family of objects in B such that, for each object X in B, the canonical arrow
x is a regular epimorphism
x:
Gj −→ X ;
j∈J,B(Gj ,X)
(AB4*) Regular epimorphisms are product-stable ;
(AB5) Filtered colimits are exact, i.e. commute with finite limits ;
(AB6) Given a small family of functors (Hi : Ai → B)I defined on small filtered categories,
the canonical comparison τ is an isomorphism
B → B) −→
(colimHi ) ;
τ : colim( Ai →
I
I
I
(WAB6) With the same notations as in (AB6), the comparison morphism τ is a regular
epimorphism.
Theory and Applications of Categories, Vol. 8, No. 17
475
It is known that
i - Condition (LM) characterizes localizations of monadic categories over a power of
Set [14] ;
ii - Conditions (LM) and (AB4*) characterize continuous localizations of monadic categories over a power of Set [1] (continuous localization means that the reflector is
continuous) ;
iii - Conditions (LM) and (AB5) characterize localizations of (multisorted, finitary) varieties [15] ;
iv - Conditions (LM), (AB4*), (AB5) and (AB6) characterize essential localizations of
varieties [1] (essential localization means that the reflector has a left adjoint).
The aim of this section is to give a new proof of the last mentioned characterization
using the theory developed in the first section, and replacing condition (AB6) with the
weaker version (WAB6).
2.1. Proposition. A complete and cocomplete category B is equivalent to an essential
localization of a variety if and only if it satisfies conditions (LM), (AB4*), (AB5) and
(WAB6).
Proof. To avoid heavy notations, let us write the proof in the one-sorted case, i.e. when
the small regular generator G is reduced to a single regular generator G. Following [15], we
consider the subcategory C of B having as objects copowers S • G of G, for S varying in
Set. An arrow G → S • G is in C if it factors through the canonical arrow S ′ • G → S • G
for some finite subset S ′ of S. There is a finitary monad over Set given by
T : Set
−•G
/ C C(G,−)/ Set
The non-full inclusion F : C → B is left covering w.r.t. weak finite limits, so that it has
an exact extension Fˆ : Cex → B. Moreover, Fˆ has a full and faithful right adjoint, so that
B is a localization of Cex . But C is equivalent to the category of free T-algebras, so that
Cex is equivalent to the variety of T-algebras. Since Cex is locally finitely presentable, B
is locally presentable [3]. Then to prove that Fˆ has a left adjoint it suffices to prove that
it is continuous [2]. According to Proposition 1.6 and Lemma 1.3, we have to show that
F is left covering w.r.t. weak small products and weak small equalizers.
I) F : C → B is left covering w.r.t. weak small products. Consider
a small family of sets
(Si )I , the corresponding objects Si • G in C and their product (πi : I (Si • G) → Si • G)I
in B. There is a canonical arrow
Σ : C(G, (Si • G)) • G −→
(Si • G)
I
where
C(G,
I
(Si • G)) = {x : G →
I
(Si • G) s.t. ∀i ∈ I x · πi is in C}.
I
Theory and Applications of Categories, Vol. 8, No. 17
476
Since a weak product in C of the Si • G’s is obtained precomposing the πi ’s with Σ, to
prove that F is left covering w.r.t.
means to prove that Σ is a regular epi.
small products
If each Si is finite, then C(G, I (Si • G)) = B(G, I (Si • G)), so that Σ is a regular epi
because G is a regular generator. In general, each Si is the filtered colimit of its finite
subsets,
Si = colimPf (Si ) Si′ , and then Si • G = colimPf (Si ) (Si′ • G) .
For each choice (Si′ ) ∈ I Pf (Si ), there is a canonical arrow
λ : C(G, (Si′ • G)) • G −→
(Si′ • G)
I
I
which is a regular epi because each Si′ is finite ; taking the colimit, we obtain a regular
epi Λ. Moreover, a diagram chase shows that there is an arrow µ making commutative
the following diagram
C(G,
Σ
I (Si • G)) • G
/
O
µ
colim I Pf (Si ) (C(G,
′
I (Si
• G)) • G)
I (Si
Λ
• G) =
′
Pf (Si ) Si
I (colim
O
• G)
τ
/ colim P (S ) (
i
I f
′
I (Si
• G))
Since Λ is a regular epi and, by condition (WAB6), τ is a regular epi, it follows that Σ is
a regular epi.
II) F : C → B is left covering w.r.t. weak small equalizers. Consider a small family of
parallel arrows in C
S•G
fi
:
/R•G
I
with S, R ∈ Set, and their equalizer e : E → S • G in B. There is a canonical arrow
Σ : C(G, e) • G → E, where
C(G, e) = {x : G → E s.t. x · e is in C}
Since a weak equalizer in C of the fi ’s is given by Σ · e : C(G, e) • G → S • G, to prove that
F is left covering w.r.t. small equalizers means to prove that Σ is a regular epi. If S is
finite, then C(G, e) = B(G, E) so that Σ is a regular epi because G is a regular generator.
In general,
S = colimPf (S) S ′ ,
and then
σS ′ : S ′ • G → S • G = colimPf (S) (S ′ • G)
is a filtered colimit. For each S ′ ∈ Pf (S), consider the following small equalizer
ES ′
eS ′
/ S′ • G
σS ′ ·fi
/
:
R•G ;
there is a canonical arrow λS ′ : C(G, eS ′ ) • G → ES ′ which is a regular epi because S ′ is
finite. Taking the colimit, we obtain a regular epi Λ. Moreover, the various σS ′ ’s give
Theory and Applications of Categories, Vol. 8, No. 17
477
rise to a canonical arrow η, and a diagram chase shows that there is an arrow µ making
commutative the following diagram
Σ
C(G, e) • G
/E
O
O
µ
η
colimPf (S) (C(G, eS ′ ) • G)
Λ
/ colimP (S) ES ′
f
To end the proof, we will show that η is an isomorphism. For this, we apply to the
diagram
fi
/
S•G : R•G I
the usual formula to compute a limit as an equalizer of a pair of arrows between two small
products, and we obtain
α
/S•G×R•G
E
β
We do the same with the diagram
S′ • G
/
/ I (R • G) × S • G × R • G
σS ′ ·fi
/
:
R•G
and we obtain
ES ′
ǫS ′
aS ′
/ S′ • G × R • G
bS ′
/
/
I (R
I
• G) × S ′ • G × R • G
Taking the colimit, we obtain the following commutative diagram
LO YYYYYYYYYY
m
l
colimPf (S) ES ′
YYYYYY
YYYYnYY
YYYYYY
YYYYYY
YYYYY,
η
/
E
ǫ
&
colimPf (S) (S ′ • G × R • G)
a
/S•G×R•G
γ
α
b
colimPf (S) ( I (R • G) × S ′ • G × R • G)
δ
/
I (R
β
• G) × S • G × R • G
where l is the equalizer of a and b. Since in B filtered colimits commute with finite limits,
the arrows m, γ and δ are isomorphisms, and then also n is an isomorphism. This implies
that η is an isomorphism.
Theory and Applications of Categories, Vol. 8, No. 17
478
3. Essential localizations of presheaf categories
In [10, 11] and [1], the following theorem is proved : a category is equivalent to an essential
localization of a presheaf category if and only if it satisfies conditions (LM) and (AB4*),
it is infinitary extensive [4] (i.e. small sums are disjoint and universal) and the following
complete distributivity holds
(CD) let I be a small set and, for each i ∈ I, consider a small family (Ai,j )j∈Ji of objects.
The canonical comparison τ is an isomorphism
( Ai,f (i) ) −→
(
Ai,j ) .
τ:
f∈
Ji i∈I
i∈I j∈Ji
Similarly to what we have done for varieties, the aim of this section is to prove the
above characterization replacing the last condition by its weaker version (WCD) in which
the comparison τ is assumed to be a regular epimorphism.
3.1. Proposition. A complete and cocomplete category B is equivalent to an essential
localization of a presheaf category if and only if it is infinitary extensive and satisfies
conditions (LM), (AB4*) and (WCD).
Proof. Let G = (Gj )J be the small regular generator, which can be seen as a full
subcategory of B. The classical Giraud’s theorem shows that B is a localization of [G op , Set]
(see [5] for a proof of Giraud’s theorem using the (finitary) exact completion). The
presheaf category [G op , Set] is equivalent to the exact completion of F amG, the sumcompletion of G (this is because the subcategory spanned by sums of representables is a
projective cover of [G op , Set]). Moreover, F amG can be described as the following non-full
subcategory
C of B : an object of C is a sum of generators,
that
is an object of the form
G
for
f
:
S
→
J
a
map
in
Set
;
an
arrow
G
→
f (s)
f (s)
s∈S
S
R Gg(r) in C is an arrow
in B such that, for each s ∈ S, there is an rs ∈ R and an arrow Gf (s) → Gg(rs ) making
commutative the following diagram
f (s)
SG
O
σs
Gf (s)
/
g(r)
RG
O
σrs
/ Gg(r )
s
Up to the equivalence [G op , Set] ≃ Cex , the left exact reflector [G op , Set] → B is given by
the exact extension Fˆ : Cex → B of the inclusion F : C → B. As in the case of varieties,
to prove that Fˆ has a left adjoint it is enough to prove that it is continuous. For this, we
have to show that F is left covering w.r.t. weak small products and weak small equalizers.
I) F : C → B is left covering w.r.t. weak small products. Observe that, given a small
category D and a functor L : D → C, a weak limit of L in C can be computed in the
following way : let (πD : L → F (LD))D be the limit in B of L · F and consider the
Theory and Applications of Categories, Vol. 8, No. 17
479
canonical morphism
λ:
Gj −→ L
j∈J,C(Gj ,L)
where
C(Gj , L) = {f : Gj → L s.t. ∀D ∈ D f · πD is in C}
A weak limit of L is obtained precomposing the πD ’s with λ. To prove that F is left
covering w.r.t. small products then means to prove that λ is a regular epi when D is
discrete, say D = I for a set I. If, for each i ∈ I, L(i) is in G, then C(Gj , L) = B(Gj , L)
and λ is a regular epi because G is a regular generator.
In general, let us write Xi = L(i)
for i ∈ I. Each Xi is a sum of generators, say Xi = j∈Ji Ai,j with Ai,j ∈ G, and we can
consider the following diagram
a
/ F(
F ( f ∈ Ji ( i∈I Ai,f (i) ))
I Xi )
O
λ
≃
f ∈ Ji (F ( i∈I Ai,f (i) ))
c
F
X
≃
i
I
i∈I ( j∈Ji F Ai,j )
O
/
b
f ∈ Ji ( i∈I F Ai,f (i) ))
The arrow a is the image under F of the canonical morphism ( Ai,j ) → Xi in C ;
the two isomorphisms depend on the fact that sums in C are computed in B ; the arrow
b is a regular epi by condition
(WCD)
; the arrow c is a regular epi because it is sum
of regular epis, each F ( I Ai,j ) → I F Ai,j being a regular epi for the particular case
previously discussed. Finally, since the diagram is commutative, λ is a regular epi.
II) F : C → B is left covering w.r.t. weak small equalizers. Once again, the proof is very
similar to that for varieties and we omit details. Let us only point out that, imitating the
constructions done in step II of the proof of Proposition 2.1, one arrives at a diagram in
which two equalizers (of pairs of parallel arrows) are compared, and one has to prove that
they are connected by two isomorphisms. This is the case because of the universality of
sums in B (in the varietal case we used the exactness of filtered colimits).
References
[1] J. Ad´amek, J. Rosick´
y, E.M. Vitale. Algebraically exact categories and essential
localizations of varieties, J. Algebra (to appear).
[2] F. Borceux. Handbook of categorical algebra, Cambridge University Press (1994).
[3] F. Borceux, G.M. Kelly. On locales of localizations, J. Pure Appl. Algebra 46 (1987)
1–34.
[4] A. Carboni, S. Lack, R.F.C. Walters. Introduction to extensive and distributive
categories, J. Pure Appl. Algebra 84 (1993) 145–158.
Theory and Applications of Categories, Vol. 8, No. 17
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[5] A. Carboni, E.M. Vitale. Regular and exact completions, J. Pure Appl. Algebra
125 (1998) 79–116.
[6] H. Hu, W. Tholen. A note on free regular and exact completions and their infinitary generalizations, Theory and Applications of Categories 2 (1996) 113–132
(http://www.tac.mta.ca/tac/).
[7] P.T. Johnstone. Remarks on quintessential and persistent localizations, Theory
and Applications of Categories 2 (1996) 90–99 (http://www.tac.mta.ca/tac/).
[8] S. Lack, E.M. Vitale. When do completion processes give rise to extensive categories? J. Pure Appl. Algebra 159 (2001) 203–230.
[9] M.C. Pedicchio, E.M. Vitale. On the abstract characterization of quasi-varieties,
Algebra Universalis 43 (2000) 269–278.
[10] J.E. Roos. Introduction a` l’´etude de la distributivit´e des foncteurs lim par rapport
aux colim dans les cat´egories des faisceaux (topos), C.R. Acad. Sci. Paris 259 (1964)
969–972.
[11] J.E. Roos. Sur la distributivit´e des foncteurs lim par rapport aux colim dans les
cat´egories des faisceaux (topos), C.R. Acad. Sci. Paris 259 (1964) 1605–1608.
[12] J.E. Roos. Caract´erisation des cat´egories qui sont quotients des cat´egories des
modules par des sous-cat´egories bilocalisantes, C.R. Acad. Sci. Paris 261 (1965)
4954–4957.
[13] J.E. Roos. Sur la condition AB6 et ses variantes dans les cat´egories ab´eliennes,
C.R. Acad. Sci. Paris 264 (1967) 991–994.
[14] E.M. Vitale. Localizations of algebraic categories, J. Pure Appl. Algebra 108 (1996)
315–320.
[15] E.M. Vitale. Localizations of algebraic categories II, J. Pure Appl. Algebra 133
(1998) 317–326.
D´epartement de Math´ematique, Universit´e catholique de Louvain,
Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium
Email: vitale@agel.ucl.ac.be
This article may be accessed via WWW at http://www.tac.mta.ca/tac/ or by anonymous ftp at ftp://ftp.tac.mta.ca/pub/tac/html/volumes/8/n17/n17.{dvi,ps}
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ESSENTIAL LOCALIZATIONS AND INFINITARY EXACT
COMPLETION
ENRICO M. VITALE
ABSTRACT. We prove the universal property of the infinitary exact completion of a
category with weak small limits. As an application, we slightly weaken the conditions
characterizing essential localizations of varieties (in particular, of module categories) and
of presheaf categories.
Introduction
An essential localization is a reflective subcategory such that the reflector has a left
adjoint. In [12], Roos gave an abstract characterization of essential localizations of module
categories, proving that they are those complete and cocomplete abelian categories with
a regular generator, satisfying the following conditions (we write the conditions in a nonabelian style, more convenient for the general framework of this work)
(AB4*) Regular epimorphisms are product-stable ;
(AB5) Filtered colimits are exact, i.e. commute with finite limits ;
(AB6) Given a small family of functors (Hi : Ai → A)I defined on small filtered categories,
the canonical comparison τ is an isomorphism
τ : colim( Ai →
A → A) −→
(colimHi ) .
I
I
I
In a subsequent paper [13], Roos introduced a weaker form of (AB6) :
(WAB6) With the same notations as in (AB6), the comparison morphism τ is a regular
epimorphism.
It is then natural to look for a representation of the abelian categories satisfying the same
list of conditions as in Roos’s theorem, but replacing (AB6) with (WAB6).
The first aim of this paper is to prove that this set of conditions, which seems weaker,
still characterizes essential localizations of module categories. In fact we prove a more
general result : Roos’s theorem has recently been generalized by Ad´amek, Rosick´
y and
the author to a non-additive context [1]. They have characterized essential localizations of
(multi-sorted, finitary) varieties as those complete and cocomplete Barr-exact categories
Received by the editors 2000 November 7 and, in revised form, 2001 August 24.
Transmitted by W. Tholen. Published on 2001 September 5.
2000 Mathematics Subject Classification: 18A35, 18B15, 18C05, 18E10, 18E35, 18F20.
c Enrico M. Vitale, 2001. Permission to copy for private use granted.
465
Theory and Applications of Categories, Vol. 8, No. 17
466
with a small regular generator, satisfying (AB4*), (AB5) and (AB6). We show that in this
characterization (AB6) can be replaced by (WAB6). Since abelian varieties are exactly
module categories, as a particular case we obtain the new formulation of Roos’s theorem.
A similar analysis can be done for Roos’s characterization of essential Grothendieck
topoi (i.e. essential localizations of presheaf categories) [10, 11] (see also Section 4 in
[1] for a different approach). In fact we prove that the complete distributivity involved in
Roos’s theorem can be weakened : instead of an isomorphism, the comparison morphism
can be assumed to be a regular epimorphism.
To obtain our results, we use in a systematic way the exact completion of a category
with weak limits [5, 6]. This uniform approach stresses one more time the striking analogy
between the varietal context and the presheaf context: not only the results are similar,
but also the proof technique is essentially the same.
In dealing with essential localizations, we need the infinitary extension of the exact
completion. This has been studied in [6]. Unfortunately, the proof of the main theorem in
[6] is not complete. The claim that a left exact functor which is left covering with respect
to small products is continuous, used to end the proof of Theorem 3.4, is not true, as we
show with a counterexample. The second aim of this paper is then to give a complete
proof of the universal property of the infinitary exact completion.
I would like to thank P.T. Johnstone for his help concerning Example 1.8.
1. The infinitary exact completion
For basic definitions and results on regular and exact categories, the reader can see [2, 5].
We assume familiarity with the exact completion as described in [5] and with the theory
of left covering functors developed therein. Left covering functors are called flat functors
in [6]. We write
Γ : C → Cex
for the exact completion of a weakly left exact category C. As far as the infinitary completion is concerned, we will deal only with arbitrary small limits ; in fact the ranked case
runs exactly as the general case.
A main difference between limits and weak limits is that in the weak case the usual
reduction to small products and equalizers of pairs of parallel arrows does not work. One
has to use small products and equalizers of small families of parallel arrows (we call them
small equalizers). The next three lemmas are the infinitary version of Proposition 1,
Lemma 28 and Proposition 27 in [5].
1.1. Lemma. If a category C has weak small products and weak small equalizers, then it
has weak small limits.
Proof. The proof is similar to that of the finitary case. We give some details in view of
Lemma 1.3.
I) Consider a small diagram of arrows with the same codomain (fi : Xi → X)I , a weak
Theory and Applications of Categories, Vol. 8, No. 17
product (πi :
I
467
Xi → Xi )I and a weak small equalizer
e
E
/
πi ·fi
:
I Xi
/X ;
then the cone (e · πi : E → Xi )I is a weak limit over (fi : Xi → X)I (a weak small
pullback).
II) Let
L : D → C be a functor with D a small category ; consider a weak product
(πD : D L(D) → L(D))D and, for each arrow d : D → D′ in D, a weak equalizer
Ed
ed
πD′
/
/
D L(D)
πD ·L(d)
′
/ L(D ) ;
consider also a weak small pullback (ǫd : E → Ed )d on the diagram (ed : Ed →
Then (ǫd · ed · πD : E → L(D))D is a weak limit of L : D → C.
D
L(D))d .
Recall that a category B is completely regular if it is regular, complete and regular
epimorphisms are stable under small products (a condition which is redundant in the
finitary case).
1.2. Lemma. Let B be a completely regular category.
1) If in the following commutative diagram, where the horizontal arrows represent small
equalizers, f1 is a regular epimorphism and f2 is a mono, then the unique factorization f is a regular epimorphism
E
e
/ A1
ai
:
/ A2
f1
f
L
/ B1
l
f2
bi
:
/ B2
2) Consider two naturally connected small families of arrows with common codomain
(that is ai · f = fi · bi for each i in a small set I)
Ai
ai
/A
fi
f
Bi
bi
/B
and the unique factorization ϕ : P → Q between the corresponding small pullbacks ;
if f is a mono and each fi is a regular epimorphism, then ϕ is a regular epimorphism.
1.3. Lemma. Let C be a category with weak small limits and B a completely regular
category. If a functor F : C → B is left covering w.r.t. weak small products and weak
small equalizers, then it is left covering w.r.t. weak small limits.
Theory and Applications of Categories, Vol. 8, No. 17
468
Proof. I) F is left covering w.r.t. weak small pullbacks : consider a small family of
convergent arrows in C and a weak pullback as described in the first step of Lemma 1.1
e
E
/
Applying F we obtain
Fe
F (E)D
L
DD
DD
t DDD
"
zS
q zzz
z
zz
|z
z
/ F(
:
uu
uu
u
u
uu s
uu
/
πi
I Xi
/ Xi
fi
/X
F (πi ·fi )
I Xi )
/ F (X)
p
1
I F (Xi )
pi
/ F (Xi )
F fi
/ F (X)
where s is the equalizer of the F (πi ·fi )’s and the bottom line is the pullback of the F (fi )’s
computed using products and equalizers in B. By assumption, p is a regular epi, so that
by Lemma 1.2 also q is a regular epi. Also t is a regular epi by assumption, so that the
comparison t · q : F (E) → L is a regular epi.
II) F is left covering w.r.t. weak small limits : consider a functor L : D → C with D
small, and a weak limit as in the second step of Lemma 1.1
ǫd
πD
/ Ed ed /
/ L(D) .
E
D L(D)
In the same way we build up the limit of L · F : D → C → B :
L
λd
/ Ld ld
/
D F (LD)
π
˜D
/ F (LD) .
Let us call l = λd · ld : L → D F (LD) ; it is the limit of
π
˜D ·F (Ld)
/
′
/ F (LD )
D F (LD)
π
˜D′
d
and then it is a mono. Now consider the pullback
l /
LO
D F O(LD)
p
p′
P
l′
/ F(
D LD)
′
By assumption,
p is a regular epi and then also p is a regular epi. Moreover, the mono
′
l : P → F ( D LD) is the limit of
F (πD )·F (Ld)
/
′
F ( D LD)
/ F (LD )
F (πD′ )
d
Theory and Applications of Categories, Vol. 8, No. 17
469
But this limit can be obtained in two steps : first taking for each d the equalizer
Sd
sd
F (πD )·F (Ld)
/ F(
D LD)
F (πD′ )
/
′
/ F (LD )
andthen taking the pullback (σd : P → Sd )d of the family of convergent arrows (sd : Sd →
F ( D LD))d . By assumption, the factorization qd : F (Ed ) → Sd is a regular epi ; then,
by Lemma 1.2, the factorization q : Q → P is a regular epi,
where (jd : Q → F (Ed ))d
is the pullback of the convergent family (F (ed ) : F (Ed ) → F ( D LD))d . Moreover, by
the first part of this lemma, the factorization t : F (E) → Q is a regular epi. Finally, the
comparison
F (E)
t
q
/Q
p′
/P
/L
is a regular epi because each part is a regular epi.
The fact that, if C has weak small limits, then Cex is completely regular, is stated
without proof in [5]. In the next lemma, we add an explicit description of small products
in Cex . This description will be used in the proof of Proposition 1.6.
1.4. Lemma. Let C be a category with weak small limits. Its exact completion Cex is
completely regular and Γ : C → Cex is left covering w.r.t. weak small limits.
Proof. The last fact follows from the fact that C is a projective cover of the complete
category Cex . The fact that in Cex regular epis are product-stable follows from the fact
that they are of the form
/S
R
X
1
/X
and from the description of products in Cex given hereunder.
Consider a family of objects
r0i
in Cex ; their product is given by
Ri
r1i
ei
E
e0
e1
I Xi πi
/
/ Xi
I
/ Ri
r0i
/ Xi
r1i
Theory and Applications of Categories, Vol. 8, No. 17
470
where πi : I Xi → Xi is a weak product in C and E is a weak limit in C as in the
following diagram
E
x FF
xx
xx
x
x| x
e0
I Xi
ei
πi
Xi o
FF e1
FF
FF
"
Ri
r0i
I Xi
πi
/ Xi
r1i
The next lemma generalizes Proposition 20 in [5]. Since the proof runs as in the
finitary case, we omit it.
1.5. Lemma. Let F : A → B be a functor between complete categories. If F is left
covering w.r.t. small limits, then it is continuous.
The next proposition is the main result of this section. It completes the preparatory
work to establish the universal property of the infinitary exact completion. In the proof,
we give an explicit description of small equalizers in Cex .
1.6. Proposition. Let C be a category with weak small limits, B an exact and completely
regular category and F : C → B a functor left covering w.r.t. weak small limits. Then the
exact extension Fˆ : Cex → B is continuous.
Proof. Recall that Fˆ sends an object of Cex , that is a pseudo equivalence relation
/
r0 , r 1 : R
/ X , to the coequalizer of the pair (F r0 , F r1 ). This coequalizer exists because, by Theorem 26 in [5], the jointly monic part of the regular epi-jointly mono factorization of (F r0 , F r1 ) is an equivalence relation. Fˆ extends to morphisms via the universal
property of the coequalizer. By Lemma 1.5, to prove that Fˆ is continuous is enough to
prove that Fˆ is left covering w.r.t. small limits. By Lemma 1.3, it suffices to prove that
Fˆ is left covering w.r.t. small products and small equalizers.
I) Consider a small product in Cex
ei
E
e0
/ Ri
r1i
r0i
e1
I Xi πi
/ Xi
Using F and Fˆ we obtain the following commutative diagram
F ( I Xi )
λ
q
/
Fˆ (e0 , e1 )
µ
/
I F (Xi )
I qi
ˆ i i
I F (r0 , r1 )
Theory and Applications of Categories, Vol. 8, No. 17
471
Each qi is a regular epi, so that I qi is a regular epi because B is completely regular ; by
assumption λ is a regular epi, so that also µ is a regular epi.
II) Small equalizers in Cex : let
fi
:
R
r0
s0
r1
X
/S
s1
fi
:
/Y
be a small family of parallel arrows in Cex and consider the diagram which contains all
the diagrams of the form
~~
~~
~
~ ~
S
s0
Y
~>
~~
~
~~
~~
s1
fi
X
/Y
fj
for each i = j, all the subdiagrams being connected by the identity on X. Take now a
weak limit E of such diagram
ϕi,j
E
s1
Y
~>
~
~~
~~
~~
e
/S
~
~
s0 ~
~~
~~
fi
X
/Y
fj
and a weak limit R as in the diagram below
e0
Eo
e1
R
e
/E
e
e
Xo
R
r0
/X
r1
The equalizer in Cex is
e
R
e0
r0
e1
E
/R
r1
e
/X
Theory and Applications of Categories, Vol. 8, No. 17
472
III) Consider a small equalizer in Cex as in the previous step
e
R
e0
r0
e1
E
fi
:
/R
e
s0
r1
/X
/S
fi
:
s1
/Y
and apply Fˆ : Cex → B. We obtain
FR
F e0
Fe
F e1
F r1
F r0
FE
F fi
:
/ FR
F fi
:
/ FX
Fe
/ FS
KK
KK p
K
F s1 KKK
F s0
K
n0 K%
o
/ FY
N (q3 )
o
n1
q2
q1
q3
ˆ
/ Fˆ (r , r ) F [fi ] / Fˆ (s , s )
0 1
0 1
:
q8
q
q
t
qqq
qqqq h
Fˆ (e0 , e1 )
Fˆ [e]
L
where the triangle on the right is the regular epi-jointly mono factorization of (F s0 , F s1 )
and the triangle on the bottom is the factorization of Fˆ [e] through the equalizer of the
Fˆ [fi ]’s. We have to prove that t : Fˆ (e0 , e1 ) → L is a regular epi. For this, consider the
limit A in B of the following diagram (a subdiagram for each i = j, all the subdiagrams
connected by the identity of F (X))
ki,j
A
i
/ N (q3 )
w
n0 ww w
w
ww
w{ w
n1
FY
x;
xx
x
xx
xx
F fi
FX
F fj
/ FY
Using the morphisms ϕi,j : E → S, we obtain a factorization σ : F E → A making commutative the following (solid) diagram
σ
FE
/A
q1
i
Fˆ (e0 , e1 )
τ
FX
q2
t
L
h
/ Fˆ (r , r )
0 1
Theory and Applications of Categories, Vol. 8, No. 17
473
Observe that h is a mono, and assume that σ is a regular epi (we will prove this later),
so that there exists τ : A → L making commutative the two triangles. If we can show
that τ is a regular epi, we have that also t is a regular epi, as we need. To show that τ is
a regular epi, we can check that it is the pullback of q3 along h. This can be done using
that (n0 , n1 ) is the kernel pair of its coequalizer q3 (because B is exact) and i is a mono
(because (n0 , n1 ) are jointly monic). It remains to prove that σ : F E → A is a regular epi.
For this, consider the limit A in B of the following diagram (with the usual conventions)
Ki,j
A
i
FY
x;
xx
x
xx
xx
/ FS
y
y
F s0 yy
y
yy
|yy
F s1
F fi
FX
F fj
/ FY
Since F : C → B is left covering, the unique factorization α : F E → A such that α · i = F e
and α · Ki,j = F ϕi,j is a regular epi. Moreover, comparing the construction of A and of A,
we see that the unique factorization m : A → A such that m · i = i and m · ki,j = Ki,j · p, is
a regular epi (use part 2 of Lemma 1.2). Finally, composing with the mono i : A → F X,
we check that σ = α · m.
1.7. Corollary. Let C be a category with weak small limits and consider its exact
completion
Γ : C → Cex .
For each exact and completely regular category B, the composition with Γ induces an
equivalence between the category of exact and continuous functors from Cex to B and the
category of functors from C to B left covering w.r.t. small limits.
Bearing in mind the fact that a left exact functor which preserves small products is
continuous, and that a functor between complete categories is left covering w.r.t. small
limits iff it is continuous (Lemma 1.5), one could think that a left exact functor which is
left covering w.r.t. small products is indeed continuous. This is wrong. In fact, let C and
B be as in Corollary 1.7 and assume F : C → B be left covering w.r.t. weak finite limits
and weak small products. A glance at the proof of Proposition 1.6 shows that the exact
extension Fˆ : Cex → B is (left exact and) left covering w.r.t. small products. Now, if Fˆ is
continuous, then F is left covering w.r.t. weak small limits. The following example shows
that this is not always true.
1.8. Example. Let C be a small category and let i : B → [C op , Set] be a persistent
localization of the presheaf category [C op , Set], i.e. the left adjoint F : [C op , Set] → B is left
exact and units are regular epis (see [7]). First of all, observe that B, being a Grothendick
topos, is exact and extensive and has a family of regular generators. Moreover, since units
are regular epis, the generators are regular projective and indecomposable objects (see
Theory and Applications of Categories, Vol. 8, No. 17
474
[9]). Then B is equivalent to a presheaf category and then (as any exact and complete
category with enough regular projective objects) it is completely regular. Let us now
prove that the reflector is left covering w.r.t. small products. Consider a family of objects
op
in [C op , Set]
(Xj )J
and their product in [C , Set] and in B. Consider also the comparison
e : F ( J Xj ) → J F (Xj ) ; we have to show that e is a regular epi. Consider the following
commutative diagram
i(e)
i(F ( J Xj ))
fMMM
MMM
M
η J Xj MMM
J Xj
/ i(
J F (X5 j ))
≃
k
kkkk
k
k
k
kkkk J ηXj
kkkk
J i(F (Xj ))
(where ηX : X → i(F (X)) is the unit at the point X). Since units are regular epis and
[C op , Set] is completely regular, i(e) is a regular epi. But, since i is full and faithful and left
exact, it reflects regular epis, and then e is a regular epi. To finish the argument, we need
an example of a persistent localization of a presheaf category such that the reflector is not
continuous. Such an example is provided by Example 2.6 in [7] : let M be a semilattice
with a non-principal ideal I and take as B the full subcategory of the category M -Set of
those objects on which the elements of I act trivially. B is a persistent localization, but
the reflector does not preserve small products, as communicated to me by P.T. Johnstone.
2. Essential localizations of varieties
Let B be a complete and cocomplete category and consider the following conditions:
(LM) B is exact and has a small regular generator G. This means that G = (Gj )J is a
small family of objects in B such that, for each object X in B, the canonical arrow
x is a regular epimorphism
x:
Gj −→ X ;
j∈J,B(Gj ,X)
(AB4*) Regular epimorphisms are product-stable ;
(AB5) Filtered colimits are exact, i.e. commute with finite limits ;
(AB6) Given a small family of functors (Hi : Ai → B)I defined on small filtered categories,
the canonical comparison τ is an isomorphism
B → B) −→
(colimHi ) ;
τ : colim( Ai →
I
I
I
(WAB6) With the same notations as in (AB6), the comparison morphism τ is a regular
epimorphism.
Theory and Applications of Categories, Vol. 8, No. 17
475
It is known that
i - Condition (LM) characterizes localizations of monadic categories over a power of
Set [14] ;
ii - Conditions (LM) and (AB4*) characterize continuous localizations of monadic categories over a power of Set [1] (continuous localization means that the reflector is
continuous) ;
iii - Conditions (LM) and (AB5) characterize localizations of (multisorted, finitary) varieties [15] ;
iv - Conditions (LM), (AB4*), (AB5) and (AB6) characterize essential localizations of
varieties [1] (essential localization means that the reflector has a left adjoint).
The aim of this section is to give a new proof of the last mentioned characterization
using the theory developed in the first section, and replacing condition (AB6) with the
weaker version (WAB6).
2.1. Proposition. A complete and cocomplete category B is equivalent to an essential
localization of a variety if and only if it satisfies conditions (LM), (AB4*), (AB5) and
(WAB6).
Proof. To avoid heavy notations, let us write the proof in the one-sorted case, i.e. when
the small regular generator G is reduced to a single regular generator G. Following [15], we
consider the subcategory C of B having as objects copowers S • G of G, for S varying in
Set. An arrow G → S • G is in C if it factors through the canonical arrow S ′ • G → S • G
for some finite subset S ′ of S. There is a finitary monad over Set given by
T : Set
−•G
/ C C(G,−)/ Set
The non-full inclusion F : C → B is left covering w.r.t. weak finite limits, so that it has
an exact extension Fˆ : Cex → B. Moreover, Fˆ has a full and faithful right adjoint, so that
B is a localization of Cex . But C is equivalent to the category of free T-algebras, so that
Cex is equivalent to the variety of T-algebras. Since Cex is locally finitely presentable, B
is locally presentable [3]. Then to prove that Fˆ has a left adjoint it suffices to prove that
it is continuous [2]. According to Proposition 1.6 and Lemma 1.3, we have to show that
F is left covering w.r.t. weak small products and weak small equalizers.
I) F : C → B is left covering w.r.t. weak small products. Consider
a small family of sets
(Si )I , the corresponding objects Si • G in C and their product (πi : I (Si • G) → Si • G)I
in B. There is a canonical arrow
Σ : C(G, (Si • G)) • G −→
(Si • G)
I
where
C(G,
I
(Si • G)) = {x : G →
I
(Si • G) s.t. ∀i ∈ I x · πi is in C}.
I
Theory and Applications of Categories, Vol. 8, No. 17
476
Since a weak product in C of the Si • G’s is obtained precomposing the πi ’s with Σ, to
prove that F is left covering w.r.t.
means to prove that Σ is a regular epi.
small products
If each Si is finite, then C(G, I (Si • G)) = B(G, I (Si • G)), so that Σ is a regular epi
because G is a regular generator. In general, each Si is the filtered colimit of its finite
subsets,
Si = colimPf (Si ) Si′ , and then Si • G = colimPf (Si ) (Si′ • G) .
For each choice (Si′ ) ∈ I Pf (Si ), there is a canonical arrow
λ : C(G, (Si′ • G)) • G −→
(Si′ • G)
I
I
which is a regular epi because each Si′ is finite ; taking the colimit, we obtain a regular
epi Λ. Moreover, a diagram chase shows that there is an arrow µ making commutative
the following diagram
C(G,
Σ
I (Si • G)) • G
/
O
µ
colim I Pf (Si ) (C(G,
′
I (Si
• G)) • G)
I (Si
Λ
• G) =
′
Pf (Si ) Si
I (colim
O
• G)
τ
/ colim P (S ) (
i
I f
′
I (Si
• G))
Since Λ is a regular epi and, by condition (WAB6), τ is a regular epi, it follows that Σ is
a regular epi.
II) F : C → B is left covering w.r.t. weak small equalizers. Consider a small family of
parallel arrows in C
S•G
fi
:
/R•G
I
with S, R ∈ Set, and their equalizer e : E → S • G in B. There is a canonical arrow
Σ : C(G, e) • G → E, where
C(G, e) = {x : G → E s.t. x · e is in C}
Since a weak equalizer in C of the fi ’s is given by Σ · e : C(G, e) • G → S • G, to prove that
F is left covering w.r.t. small equalizers means to prove that Σ is a regular epi. If S is
finite, then C(G, e) = B(G, E) so that Σ is a regular epi because G is a regular generator.
In general,
S = colimPf (S) S ′ ,
and then
σS ′ : S ′ • G → S • G = colimPf (S) (S ′ • G)
is a filtered colimit. For each S ′ ∈ Pf (S), consider the following small equalizer
ES ′
eS ′
/ S′ • G
σS ′ ·fi
/
:
R•G ;
there is a canonical arrow λS ′ : C(G, eS ′ ) • G → ES ′ which is a regular epi because S ′ is
finite. Taking the colimit, we obtain a regular epi Λ. Moreover, the various σS ′ ’s give
Theory and Applications of Categories, Vol. 8, No. 17
477
rise to a canonical arrow η, and a diagram chase shows that there is an arrow µ making
commutative the following diagram
Σ
C(G, e) • G
/E
O
O
µ
η
colimPf (S) (C(G, eS ′ ) • G)
Λ
/ colimP (S) ES ′
f
To end the proof, we will show that η is an isomorphism. For this, we apply to the
diagram
fi
/
S•G : R•G I
the usual formula to compute a limit as an equalizer of a pair of arrows between two small
products, and we obtain
α
/S•G×R•G
E
β
We do the same with the diagram
S′ • G
/
/ I (R • G) × S • G × R • G
σS ′ ·fi
/
:
R•G
and we obtain
ES ′
ǫS ′
aS ′
/ S′ • G × R • G
bS ′
/
/
I (R
I
• G) × S ′ • G × R • G
Taking the colimit, we obtain the following commutative diagram
LO YYYYYYYYYY
m
l
colimPf (S) ES ′
YYYYYY
YYYYnYY
YYYYYY
YYYYYY
YYYYY,
η
/
E
ǫ
&
colimPf (S) (S ′ • G × R • G)
a
/S•G×R•G
γ
α
b
colimPf (S) ( I (R • G) × S ′ • G × R • G)
δ
/
I (R
β
• G) × S • G × R • G
where l is the equalizer of a and b. Since in B filtered colimits commute with finite limits,
the arrows m, γ and δ are isomorphisms, and then also n is an isomorphism. This implies
that η is an isomorphism.
Theory and Applications of Categories, Vol. 8, No. 17
478
3. Essential localizations of presheaf categories
In [10, 11] and [1], the following theorem is proved : a category is equivalent to an essential
localization of a presheaf category if and only if it satisfies conditions (LM) and (AB4*),
it is infinitary extensive [4] (i.e. small sums are disjoint and universal) and the following
complete distributivity holds
(CD) let I be a small set and, for each i ∈ I, consider a small family (Ai,j )j∈Ji of objects.
The canonical comparison τ is an isomorphism
( Ai,f (i) ) −→
(
Ai,j ) .
τ:
f∈
Ji i∈I
i∈I j∈Ji
Similarly to what we have done for varieties, the aim of this section is to prove the
above characterization replacing the last condition by its weaker version (WCD) in which
the comparison τ is assumed to be a regular epimorphism.
3.1. Proposition. A complete and cocomplete category B is equivalent to an essential
localization of a presheaf category if and only if it is infinitary extensive and satisfies
conditions (LM), (AB4*) and (WCD).
Proof. Let G = (Gj )J be the small regular generator, which can be seen as a full
subcategory of B. The classical Giraud’s theorem shows that B is a localization of [G op , Set]
(see [5] for a proof of Giraud’s theorem using the (finitary) exact completion). The
presheaf category [G op , Set] is equivalent to the exact completion of F amG, the sumcompletion of G (this is because the subcategory spanned by sums of representables is a
projective cover of [G op , Set]). Moreover, F amG can be described as the following non-full
subcategory
C of B : an object of C is a sum of generators,
that
is an object of the form
G
for
f
:
S
→
J
a
map
in
Set
;
an
arrow
G
→
f (s)
f (s)
s∈S
S
R Gg(r) in C is an arrow
in B such that, for each s ∈ S, there is an rs ∈ R and an arrow Gf (s) → Gg(rs ) making
commutative the following diagram
f (s)
SG
O
σs
Gf (s)
/
g(r)
RG
O
σrs
/ Gg(r )
s
Up to the equivalence [G op , Set] ≃ Cex , the left exact reflector [G op , Set] → B is given by
the exact extension Fˆ : Cex → B of the inclusion F : C → B. As in the case of varieties,
to prove that Fˆ has a left adjoint it is enough to prove that it is continuous. For this, we
have to show that F is left covering w.r.t. weak small products and weak small equalizers.
I) F : C → B is left covering w.r.t. weak small products. Observe that, given a small
category D and a functor L : D → C, a weak limit of L in C can be computed in the
following way : let (πD : L → F (LD))D be the limit in B of L · F and consider the
Theory and Applications of Categories, Vol. 8, No. 17
479
canonical morphism
λ:
Gj −→ L
j∈J,C(Gj ,L)
where
C(Gj , L) = {f : Gj → L s.t. ∀D ∈ D f · πD is in C}
A weak limit of L is obtained precomposing the πD ’s with λ. To prove that F is left
covering w.r.t. small products then means to prove that λ is a regular epi when D is
discrete, say D = I for a set I. If, for each i ∈ I, L(i) is in G, then C(Gj , L) = B(Gj , L)
and λ is a regular epi because G is a regular generator.
In general, let us write Xi = L(i)
for i ∈ I. Each Xi is a sum of generators, say Xi = j∈Ji Ai,j with Ai,j ∈ G, and we can
consider the following diagram
a
/ F(
F ( f ∈ Ji ( i∈I Ai,f (i) ))
I Xi )
O
λ
≃
f ∈ Ji (F ( i∈I Ai,f (i) ))
c
F
X
≃
i
I
i∈I ( j∈Ji F Ai,j )
O
/
b
f ∈ Ji ( i∈I F Ai,f (i) ))
The arrow a is the image under F of the canonical morphism ( Ai,j ) → Xi in C ;
the two isomorphisms depend on the fact that sums in C are computed in B ; the arrow
b is a regular epi by condition
(WCD)
; the arrow c is a regular epi because it is sum
of regular epis, each F ( I Ai,j ) → I F Ai,j being a regular epi for the particular case
previously discussed. Finally, since the diagram is commutative, λ is a regular epi.
II) F : C → B is left covering w.r.t. weak small equalizers. Once again, the proof is very
similar to that for varieties and we omit details. Let us only point out that, imitating the
constructions done in step II of the proof of Proposition 2.1, one arrives at a diagram in
which two equalizers (of pairs of parallel arrows) are compared, and one has to prove that
they are connected by two isomorphisms. This is the case because of the universality of
sums in B (in the varietal case we used the exactness of filtered colimits).
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D´epartement de Math´ematique, Universit´e catholique de Louvain,
Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium
Email: vitale@agel.ucl.ac.be
This article may be accessed via WWW at http://www.tac.mta.ca/tac/ or by anonymous ftp at ftp://ftp.tac.mta.ca/pub/tac/html/volumes/8/n17/n17.{dvi,ps}
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